Menu

Prove that \[\lim_{h\to 0}\,\frac{h^n}{h}=0\]where $n$ is a positive integer greater than $1$.

Primary tabs

Lesson Parent:

Problem:

Answer:

It is true that \[\lim_{h\to 0}\,\frac{h^n}{h}=0\]where $n$ is a positive integer.

Solution:

Our problem states that <Sign in to see all the formulas>. Note that we have proven this for the case when $n=2$ and $n=3$ here and here. Let's just assume that <Sign in to see all the formulas>, but the proof does not require this. It is good for <Sign in to see all the formulas>.

For this particular problem, we need to show that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that if <Sign in to see all the formulas> then <Sign in to see all the formulas>. The important thing to note is that <Sign in to see all the formulas> except when $h=0$. When $h=0$ then $h^n/h$ is undefined since we cannot divide by zero. However, our function limit proof states that we evaluate the function $h^n/h$ for values of $h$ in <Sign in to see all the formulas>. In particular, <Sign in to see all the formulas> means that we evaluate the function $h^n/h$ at values of <Sign in to see all the formulas>. Therefore, when <Sign in to see all the formulas> then <Sign in to see all the formulas>.